Representations of simple noncommutative Jordan superalgebras I

Abstract

In this article we begin the study of representations of simple finite-dimensional noncommutative Jordan superalgebras. In the case of degree ≥3 we show that any finite-dimensional representation is completely reducible and, depending on the superalgebra, quasiassociative or Jordan. Then we study representations of superalgebras Dt(α,β,γ) and K3(α,β,γ) and prove the Kronecker factorization theorem for superalgebras Dt(α,β,γ). In the last section we use a new approach to study noncommutative Jordan representations of simple Jordan superalgebras.